code: def test_inverse_laplace_transform():
      s = symbols("s")
      k, n, t = symbols("k, n, t", real=True)
      a, b, c, d = symbols("a, b, c, d", positive=True)
      f = Function("f")
      F = Function("F")
  
      def ILT(g):
          return inverse_laplace_transform(g, s, t)
  
      def ILTS(g):
          return inverse_laplace_transform(g, s, t, simplify=True)
  
      def ILTF(g):
          return laplace_correspondence(
              inverse_laplace_transform(g, s, t), {f: F}
          )
  
      # Tests for the rules in Bateman54.
  
      # Section 4.1: Some of the Laplace transform rules can also be used well
      #     in the inverse transform.
      assert ILTF(exp(-a * s) * F(s)) == f(-a + t)
      assert ILTF(k * F(s - a)) == k * f(t) * exp(-a * t)
      assert ILTF(diff(F(s), s, 3)) == -(t**3) * f(t)
      assert ILTF(diff(F(s), s, 4)) == t**4 * f(t)
  
      # Section 5.1: Most rules are impractical for a computer algebra system.
  
      # Section 5.2: Rational functions
      assert ILT(2) == 2 * DiracDelta(t)
      assert ILT(1 / s) == Heaviside(t)
      assert ILT(1 / s**2) == t * Heaviside(t)
      assert ILT(1 / s**5) == t**4 * Heaviside(t) / 24
      assert ILT(1 / s**n) == t ** (n - 1) * Heaviside(t) / gamma(n)
      assert ILT(a / (a + s)) == a * exp(-a * t) * Heaviside(t)
      assert ILT(s / (a + s)) == -a * exp(-a * t) * Heaviside(
          t
      ) + DiracDelta(t)
      assert ILT(b * s / (s + a) ** 2) == b * (
          -a * t * exp(-a * t) * Heaviside(t)
          + exp(-a * t) * Heaviside(t)
      )
      assert ILTS(c / ((s + a) * (s + b))) == c * (
          exp(a * t) - exp(b * t)
      ) * exp(-t * (a + b)) * Heaviside(t) / (a - b)
      assert ILTS(c * s / ((s + a) * (s + b))) == c * (
          a * exp(b * t) - b * exp(a * t)
      ) * exp(-t * (a + b)) * Heaviside(t) / (a - b)
      assert (
          ILTS(s / (a + s) ** 3)
          == t * (-a * t + 2) * exp(-a * t) * Heaviside(t) / 2
      )
      assert ILTS(1 / (s * (a + s) ** 3)) == (
          -(a**2) * t**2 - 2 * a * t + 2 * exp(a * t) - 2
      ) * exp(-a * t) * Heaviside(t) / (2 * a**3)
      assert ILT(1 / (s * (a + s) ** n)) == (
          Heaviside(t) * lowergamma(n, a * t) / (a**n * gamma(n))
      )
      assert ILT((s - a) ** (-b)) == t ** (b - 1) * exp(
          a * t
      ) * Heaviside(t) / gamma(b)
      assert ILT((a + s) ** (-2)) == t * exp(-a * t) * Heaviside(t)
      assert (
          ILT((a + s) ** (-5))
          == t**4 * exp(-a * t) * Heaviside(t) / 24
      )
      assert ILT(s**2 / (s**2 + 1)) == -sin(t) * Heaviside(
          t
      ) + DiracDelta(t)
      assert ILT(1 - 1 / (s**2 + 1)) == -sin(t) * Heaviside(
          t
      ) + DiracDelta(t)
      assert ILT(a / (a**2 + s**2)) == sin(a * t) * Heaviside(t)
      assert ILT(s / (s**2 + a**2)) == cos(a * t) * Heaviside(t)
      assert ILT(b / (b**2 + (a + s) ** 2)) == exp(-a * t) * sin(
          b * t
      ) * Heaviside(t)
      assert ILT(b * s / (b**2 + (a + s) ** 2)) == b * (
          -a * exp(-a * t) * sin(b * t) / b + exp(-a * t) * cos(b * t)
      ) * Heaviside(t)
      assert ILT(1 / (s**2 * (s**2 + 1))) == t * Heaviside(t) - sin(
          t
      ) * Heaviside(t)
      assert ILTS(c * s / (d**2 * (s + a) ** 2 + b**2)) == c * (
          -a * d * sin(b * t / d) + b * cos(b * t / d)
      ) * exp(-a * t) * Heaviside(t) / (b * d**2)
      assert ILTS((b * s**2 + d) / (a**2 + s**2) ** 2) == (
          2 * a**2 * b * sin(a * t)
          + (a**2 * b - d) * (a * t * cos(a * t) - sin(a * t))
      ) * Heaviside(t) / (2 * a**3)
      assert (
          ILTS(b / (s**2 - a**2))
          == b * sinh(a * t) * Heaviside(t) / a
      )
      assert ILT(b / (s**2 - a**2)) == b * (
          exp(a * t) * Heaviside(t) / (2 * a)
          - exp(-a * t) * Heaviside(t) / (2 * a)
      )
      assert ILTS(b * s / (s**2 - a**2)) == b * cosh(
          a * t
      ) * Heaviside(t)
      assert ILT(b / (s * (s + a))) == b * (
          Heaviside(t) / a - exp(-a * t) * Heaviside(t) / a
      )
      # Issue #24424
      assert (
          ILTS((s + 8) / ((s + 2) * (s**2 + 2 * s + 10)))
          == ((8 * sin(3 * t) - 9 * cos(3 * t)) * exp(t) + 9)
          * exp(-2 * t)
          * Heaviside(t)
          / 15
      )
      # Issue #8514; this is not important anymore, since this function
      # is not solved by integration anymore
      assert ILT(1 / (a * s**2 + b * s + c)) == 2 * exp(
          -b * t / (2 * a)
      ) * sin(t * sqrt(4 * a * c - b**2) / (2 * a)) * Heaviside(
          t
      ) / sqrt(
          4 * a * c - b**2
      )
  
      # Section 5.3: Irrational algebraic functions
      assert ILT(1 / sqrt(s) / (b * s - a)) == exp(  # (1)
          a * t / b
      ) * Heaviside(t) * erf(sqrt(a) * sqrt(t) / sqrt(b)) / (
          sqrt(a) * sqrt(b)
      )
      assert ILT(1 / sqrt(k * s) / (c * s - a) / s) == (  # (2)
          -2 * c * sqrt(t) / (sqrt(pi) * a)
          + c ** (S(3) / 2)
          * exp(a * t / c)
          * erf(sqrt(a) * sqrt(t) / sqrt(c))
          / a ** (S(3) / 2)
      ) * Heaviside(t) / (c * sqrt(k))
      assert ILT(1 / (sqrt(c * s) + a)) == (  # (4)
          -a * exp(a**2 * t / c) * erfc(a * sqrt(t) / sqrt(c)) / c
          + 1 / (sqrt(pi) * sqrt(c) * sqrt(t))
      ) * Heaviside(t)
      assert ILT(a / s / (b * sqrt(s) + a)) == (  # (5)
          -exp(a**2 * t / b**2) * erfc(a * sqrt(t) / b) + 1
      ) * Heaviside(t)
      assert ILT(  # (6)
          (a - b) * sqrt(s) / (sqrt(s) + sqrt(a)) / (s - b)
      ) == (
          sqrt(a) * sqrt(b) * exp(b * t) * erfc(sqrt(b) * sqrt(t))
          + a * exp(a * t) * erfc(sqrt(a) * sqrt(t))
          - b * exp(b * t)
      ) * Heaviside(
          t
      )
      assert ILT(1 / sqrt(s) / (sqrt(b * s) + a)) == exp(  # (7)
          a**2 * t / b
      ) * Heaviside(t) * erfc(a * sqrt(t) / sqrt(b)) / sqrt(b)
      assert ILT(a**2 / (sqrt(s) + a) / s ** (S(3) / 2)) == (  # (8)
          2 * a * sqrt(t) / sqrt(pi)
          + exp(a**2 * t) * erfc(a * sqrt(t))
          - 1
      ) * Heaviside(t)
      assert ILT(  # (9)
          (a - b) * sqrt(b) / (s - b) / sqrt(s) / (sqrt(s) + sqrt(a))
      ) == (
          sqrt(a) * exp(b * t) * erf(sqrt(b) * sqrt(t))
          + sqrt(b) * exp(a * t) * erfc(sqrt(a) * sqrt(t))
          - sqrt(b) * exp(b * t)
      ) * Heaviside(
          t
      )
      assert ILT(1 / (sqrt(s) + sqrt(a)) ** 2) == (  # (10)
          -2 * sqrt(a) * sqrt(t) / sqrt(pi)
          + (-2 * a * t + 1) * (erf(sqrt(a) * sqrt(t)) - 1) * exp(a * t)
          + 1
      ) * Heaviside(t)
      assert ILT(1 / (sqrt(s) + sqrt(a)) ** 2 / s) == (  # (11)
          (2 * t - 1 / a) * exp(a * t) * erfc(sqrt(a) * sqrt(t))
          + 1 / a
          - 2 * sqrt(t) / (sqrt(pi) * sqrt(a))
      ) * Heaviside(t)
      assert ILT(1 / (sqrt(s) + a) ** 2 / sqrt(s)) == (  # (12)
          -2 * a * t * exp(a**2 * t) * erfc(a * sqrt(t))
          + 2 * sqrt(t) / sqrt(pi)
      ) * Heaviside(t)
      assert ILT(1 / (sqrt(s) + a) ** 3) == (  # (13)
          -a
          * t
          * (2 * a**2 * t + 3)
          * exp(a**2 * t)
          * erfc(a * sqrt(t))
          + 2 * sqrt(t) * (a**2 * t + 1) / sqrt(pi)
      ) * Heaviside(t)
      x = (
          -ILT(sqrt(s) / (sqrt(s) + a) ** 3)
          + 2
          * (
              sqrt(pi)
              * a**2
              * t
              * (
                  -2 * sqrt(pi) * erfc(a * sqrt(t))
                  + 2 * exp(-(a**2) * t) / (a * sqrt(t))
              )
              * (-(a**4) * t**2 - 5 * a**2 * t / 2 - S.Half)
              * exp(a**2 * t)
              / 2
              + sqrt(pi) * a * sqrt(t) * (a**2 * t + 1) / 2
          )
          * Heaviside(t)
          / (pi * a**2 * t)
      ).simplify()
      assert x == 0  # (14)
      x = (
          -ILT(1 / sqrt(s) / (sqrt(s) + a) ** 3)
          + Heaviside(t)
          * (
              sqrt(t)
              * (
                  (2 * a**2 * t + 1)
                  * (
                      sqrt(pi)
                      * a
                      * sqrt(t)
                      * exp(a**2 * t)
                      * erfc(a * sqrt(t))
                      - 1
                  )
                  + 1
              )
              / (sqrt(pi) * a)
          )
      ).simplify()
      assert x == 0  # (15)
      assert factor_terms(ILT(3 / (sqrt(s) + a) ** 4)) == 3 * (  # (16)
          -2
          * a**3
          * t ** (S(5) / 2)
          * (2 * a**2 * t + 5)
          / (3 * sqrt(pi))
          + t
          * (4 * a**4 * t**2 + 12 * a**2 * t + 3)
          * exp(a**2 * t)
          * erfc(a * sqrt(t))
          / 3
      ) * Heaviside(t)
      assert ILT((sqrt(s) - a) / (s * (sqrt(s) + a))) == (  # (17)
          2 * exp(a**2 * t) * erfc(a * sqrt(t)) - 1
      ) * Heaviside(t)
      assert ILT(  # (18)
          (sqrt(s) - a) ** 2 / (s * (sqrt(s) + a) ** 2)
      ) == (
          1
          + 8 * a**2 * t * exp(a**2 * t) * erfc(a * sqrt(t))
          - 8 / sqrt(pi) * a * sqrt(t)
      ) * Heaviside(
          t
      )
      assert ILT(  # (19)
          (sqrt(s) - a) ** 3 / (s * (sqrt(s) + a) ** 3)
      ) == Heaviside(t) * (
          2
          * (8 * a**4 * t**2 + 8 * a**2 * t + 1)
          * exp(a**2 * t)
          * erfc(a * sqrt(t))
          - 8 / sqrt(pi) * a * sqrt(t) * (2 * a**2 * t + 1)
          - 1
      )
      assert ILT(sqrt(s + a) / (s + b)) == Heaviside(t) * (  # (22)
          exp(-a * t) / sqrt(t) / sqrt(pi)
          + sqrt(a - b) * exp(-b * t) * erf(sqrt(a - b) * sqrt(t))
      )
      assert ILT(1 / sqrt(s + b) / (s + a)) == Heaviside(t) * (  # (23)
          1 / sqrt(b - a) * exp(-a * t) * erf(sqrt(b - a) * sqrt(t))
      )
      assert ILT(1 / sqrt(s**2 + a**2)) == Heaviside(t) * (  # (35)
          besselj(0, a * t)
      )
      assert ILT(1 / sqrt(s**2 - a**2)) == Heaviside(t) * (  # (44)
          besseli(0, a * t)
      )
  
      # Miscellaneous tests
      # Can _inverse_laplace_time_shift deal with positive exponents?
      assert (
          -ILT(
              (s**2 * exp(2 * s) + 4 * exp(s) - 4)
              * exp(-2 * s)
              / (s * (s**2 + 1))
          )
          + cos(t) * Heaviside(t)
          + 4 * cos(t - 2) * Heaviside(t - 2)
          - 4 * cos(t - 1) * Heaviside(t - 1)
          - 4 * Heaviside(t - 2)
          + 4 * Heaviside(t - 1)
      ).simplify() == 0
  
imports: from sympy.integrals.laplace import laplace_transform, inverse_laplace_transform, LaplaceTransform, InverseLaplaceTransform, _laplace_deep_collect, laplace_correspondence, laplace_initial_conds
  from sympy.core.function import Function, expand_mul
  from sympy.core import EulerGamma, Subs, Derivative, diff
  from sympy.core.exprtools import factor_terms
  from sympy.core.numbers import I, oo, pi
  from sympy.core.relational import Eq
  from sympy.core.singleton import S
  from sympy.core.symbol import Symbol, symbols
  from sympy.simplify.simplify import simplify
  from sympy.functions.elementary.complexes import Abs, re
  from sympy.functions.elementary.exponential import exp, log, exp_polar
  from sympy.functions.elementary.hyperbolic import cosh, sinh, coth, asinh
  from sympy.functions.elementary.miscellaneous import sqrt
  from sympy.functions.elementary.piecewise import Piecewise
  from sympy.functions.elementary.trigonometric import atan, cos, sin
  from sympy.logic.boolalg import And
  from sympy.functions.special.gamma_functions import lowergamma, gamma, uppergamma
  from sympy.functions.special.delta_functions import DiracDelta, Heaviside
  from sympy.functions.special.singularity_functions import SingularityFunction
  from sympy.functions.special.zeta_functions import lerchphi
  from sympy.functions.special.error_functions import fresnelc, fresnels, erf, erfc, Ei, Ci, expint, E1
  from sympy.functions.special.bessel import besseli, besselj, besselk, bessely
  from sympy.testing.pytest import slow, warns_deprecated_sympy
  from sympy.matrices import Matrix, eye
  from sympy.abc import s
label_desc: Use image node_id 2 for calling a global function with example usage: test_inverse_laplace_transform() and returns: inverse_laplace_transform, inverse_laplace_transform, laplace_correspondence